4 1. Various Ways of Representing Surfaces and Examples Figure 1.3. A torus and a handle. Another familiar example of a surface is a torus—just as the sphere is the surface of an idealised ball, the torus is the surface of an idealised doughnut (or perhaps a bagel, depending on what sort of diet one is on). Like our first three examples, it is a surface of revolution, and may be obtained by rotating a circle around a line which lies in the plane of the circle, but does not intersect it. We will derive a nice equation (1.5) for the torus in the next lecture. We can obtain new surfaces with qualitatively distinctive shapes by the procedure called “attaching a handle”. A handle can be thought of as a torus with a hole (or if you like, an inner tube with a small patch cut out), as shown in Figure 1.3—this is attached to a hole cut in a given surface. Applying this procedure to a sphere produces a surface in the general shape of a torus. If we continue to attach more handles, we obtain something reminiscent of a pretzel with an increasing number of holes or, alternatively, a chain of tori linked to each other—Figure 1.4 shows a sphere with two handles. Like all the surfaces we have dealt with so far, these surfaces can also be represented by equations with a certain amount of effort (see Exercise 1.6). b. Equations vs. other methods. We have obtained several dif- ferent surfaces as the set of points whose coordinates (x, y, z) satisfy one equation or another. It is natural to ask what sort of equations will always yield nice, recognisable surfaces. Will any old equation do? Or must we impose some restrictions? And conversely, can we represent every surface by an equation?

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